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Convex Shape - Definition, Properties & Examples

TL;DR

A convex shape is a shape where the straight line joining any two points inside it stays completely inside - equivalently, every interior angle of a convex polygon is less than 180°. This guide gives the one-line test, the properties, six worked examples, and the mistakes that make students misclassify a shape.

What Is A Convex Shape?

A convex shape is a shape in which, for every pair of points you pick inside it, the entire straight segment joining them lies inside the shape. Nothing pokes inward. For a convex polygon - a closed figure made of straight sides - this is the same as saying every interior angle is less than 180°.

The opposite is a concave shape (also called non-convex), which has at least one interior angle greater than 180°, so it looks like it has a "bite" taken out of it. A line drawn between two points inside a concave shape can pass outside the figure.

The quick test: pick any two points inside the shape and imagine the straight line between them. If that line can ever leave the shape, the shape is concave; if it can never leave, the shape is convex.

Examples Of Convex Shape

These examples build from naming shapes to testing them with the angle rule and the segment rule. Each problem statement is bold; the steps are plain.

Example 1

Is an equilateral triangle a convex shape?

Every triangle has three interior angles that add to 180°, so no single angle can reach 180°. Each angle in an equilateral triangle is 60°.

Since every angle is under 180°, the triangle is convex.

Final answer: yes, every triangle is convex.

Example 2

A quadrilateral has interior angles 80°, 200°, 30°, and 50°. Is it convex?

Your first instinct is to check that the angles add to 360° (the angle sum of any quadrilateral) and, seeing 80 + 200 + 30 + 50=360, to declare the shape a valid convex quadrilateral. Let's look closer.

The angle sum being correct only confirms it is a genuine quadrilateral. It says nothing about convexity. Scan the individual angles: 200° is greater than 180°.

That single reflex angle means the shape caves in at one vertex, so a segment across it would leave the figure.

Final answer: no, it is concave, because one interior angle (200°) exceeds 180°.

Example 3

Is a circle a convex shape?

A circle is not a polygon, so use the segment test instead of the angle rule. Pick any two points inside a circle and draw the segment between them.

That segment always stays inside the disc; it never bulges out through the curve.

Final answer: yes, a circle (and its filled disc) is convex.

Example 4

Is a regular hexagon convex? Use the interior-angle formula.

The interior angle of a regular polygon with n sides is ( \frac{(n-2) \times 180°}{n} ). For a hexagon, n=6:

[ \frac{(6-2) \times 180°}{6} = \frac{720°}{6} = 120° ]

Every angle is 120°, which is under 180°.

Final answer: yes, a regular hexagon is convex.

Example 5

Which of these are convex: a square, a crescent moon shape, an arrow (chevron)?

• A square has four 90° angles, all under 180° - convex.
• A crescent has a curved inward bite, so a segment across the horns leaves the shape - concave.
• An arrow / chevron has a notch at the tail, giving one reflex angle - concave.

Final answer: only the square is convex.

Example 6

A polygon has all its diagonals lying inside it. Is it convex or concave?

For a convex polygon, every diagonal lies inside the figure. For a concave polygon, at least one diagonal passes outside.

Since all diagonals lie inside, the shape passes the convex test.

Final answer: it is convex - the "all diagonals inside" property is another way to state convexity.

Why Convexity Matters: "The Tightest Skin Around a Set of Points"

Convexity is one of the oldest useful ideas in geometry because it captures "no dents, no surprises." When you stretch a rubber band around a scattering of nails, it snaps to the convex hull - the smallest convex shape containing them all. That single idea drives a lot of real work.

• Structural safety. A convex cross-section spreads load smoothly; a concave notch concentrates stress at the inward corner, which is exactly where cracks start.
• Optics and dishes. A convex lens or a satellite dish curves outward so that rays converge predictably, with no inward fold to scatter them.
• Algorithms and graphics. Collision detection, packaging, and route-planning lean on convex regions because a straight line between two inside points never escapes - which makes "is it inside?" cheap to check.

Common Mistakes With Convex Shapes

These errors show up the moment a student has to classify a shape under time pressure.

Mistake 1: Judging convexity from a "rounded" look

Where it slips in: Assuming any curvy or smooth-looking shape must be convex.

Don't do this: Calling a crescent moon convex because its edges are smooth curves.

The correct way: Smoothness is not convexity. A crescent has an inward-curving edge, so a segment between its two horns leaves the shape. Apply the segment test, not a gut feeling about how "round" it looks.

Mistake 2: Checking only the angle sum, not each angle

Where it slips in: Confirming the interior angles add to the right total and stopping there.

Don't do this: Seeing a quadrilateral's angles sum to 360° and concluding it is convex.

The correct way: The angle sum only confirms the shape is a valid polygon. Convexity needs every individual angle to be under 180°. Scan for any single reflex angle. The second-guesser who trusts the total without inspecting each angle will pass concave shapes as convex.

Mistake 3: Forgetting non-polygon shapes need the segment test

Where it slips in: Trying to use the "<180° angle" rule on a shape with curved sides.

Don't do this: Looking for interior angles on a circle or a blob and getting stuck.

The correct way: The angle rule is only for polygons. For any shape - curved or straight - the universal test is the segment rule: can a line between two inside points ever leave the shape? If never, it is convex.

Conclusion

• A convex shape is one where every line between two inside points stays inside the shape.
• For a convex polygon, this is the same as every interior angle being less than 180°.
• A concave shape has at least one reflex angle (over 180°) and looks like it has a bite taken out.
• All diagonals of a convex polygon lie inside it; some diagonals of a concave polygon lie outside.
• Use the angle rule for polygons and the segment test for any shape, including curved ones.

Practise What You Have Learned

Work through these to test your understanding: decide whether a regular octagon is convex using the interior-angle formula; classify a shape with angles 100°, 95°, 185°, 80° (Answer to Question 2: concave, because 185°>180°); and sketch one convex and one concave hexagon.